1. Field of the Invention
The present invention relates to methods for generating free curves and sculptured surfaces for use with the designing apparatus for CAD/CAM (computer aided design/computer aided manufacturing) applications.
2. Description of the Prior Art
In a geometric modeling application where an object with a sculptured surface is to be designed, the designer designates a plurality of points (called nodes) in the three-dimensional space to be traversed by that surface. The designated nodes are calculated using a desired vector function so that the nodes are connected by a boundary curve network. This provides a surface expressed by what is known as a wire frame. In this manner, there are generated a large number of frame spaces enclosed by boundary curves (this is called a framing process).
The boundary curve network generated by the framing process presents itself approximately as what the designer wants to design. In this case, a surface may be interpolated as the expression of a predetermined vector function using the boundary curves that enclose each frame space. Taken as a whole, this is the sculptured surface designed by the designer (and not defined by secondary functions). Each surface occupying each frame space constitutes a component that makes up the entire surface. The component is called a patch.
There was proposed a free curve generating method whereby the control side vectors around a common boundary between two adjacent frame spaces are modified in order to establish a patch that will satisfy the condition for tangent plane continuation on that common boundary. The major object of this prior art method is to provide the entire sculptured surface with a more natural external shape.
As shown in FIG. 27, this prior art sculptured surface generating method involves first locating a patch vector S(u, v)1 and a patch vector S(u, v)2 in a rectangular frame space and having them expressed by a vector function S(u, v) constituting a cubic Bezier equation. The two patch vectors S(u, v)1 and S(u, v)2 are connected smoothly as follows. When node vectors P(00), P(30)1, P(33)1, P(03), P(33)2 and P(30)2 are furnished by the framing process, there are accordingly established control side vectors a1, a2, c1 and c2 in such a manner that the condition for tangent plane continuation is met on a common boundary COM between adjacent patch vectors S(u, v)1 and S(u, v)2. These control side vectors are used to modify control point vectors P(11)1, P(12)1, P(11)2 and P(12)2.
The above method is applied consecutively to other common boundaries. This eventually results in the smooth connection of the patch vectors S(u, v)1 and S(u, v)2 on the condition of tangent plane continuation with the adjacent patches.
The vector function S(u, v) constituted by a cubic Bezier equation is expressed as EQU S(u,v)=(1-u+u E).sup.3 (1-v+v F).sup.3 P(00) (1)
where, u and v are parameters for u and v directions respectively, and E and F are shift operators. Furthermore, the vector function S(u, v) has those relations with a control point vector P(ij) which are defined as EQU E.multidot.P(ij)=P(i+1 j) (i j=0 1 2) (2) EQU F.multidot.P(ij)=P(i j+1) (i j=0 1 2) (3) EQU 0.gtoreq.u.gtoreq.1 (4) EQU 0.gtoreq.v.gtoreq.1 (5)
The tangent plane is a plane formed by tangent vectors in the u and v directions at each of the nodes on a common boundary. For example, where the patch vectors S(u, v)1 and S(u, v)2 in FIG. 27 share the same tangent plane on a common boundary COM12, the condition of tangent plane continuation is met.
The above prior art method allows the designer to design cubic shapes that are otherwise difficult to design, such as those whose surfaces change smoothly. Nevertheless, some improvements may be envisaged for this method.
On a designing apparatus operating on the above method, it will be convenient if the external shape of a target object is input in the form of a sequence of points. This requires forming a wire frame model that connects the sequence of the points that were input. It is also necessary to have Bezier curves represent the free curves that constitute the wire frame model. Furthermore, on the generated wire frame model, a sculptured surface must be formed so that it will traverse the input points.
If the patches generated in the above process are corrected using curved or otherwise shaped surfaces, it will also be convenient. The ability to modify the surface shape in this manner will make it possible not only to correct patch size but also to set up the frame space anew as needed.